Simpler (classical) and faster (quantum) algorithms for Gibbs partition functions

TL;DR

This paper introduces improved classical and quantum algorithms for approximating partition functions of classical Hamiltonians, reducing sample complexity and cooling schedule length, and achieving quadratic speedups over previous methods.

Contribution

It modifies the classical algorithm of fefankovid, Vempala, and Vigoda to improve sample complexity and quantizes it, surpassing prior quantum algorithms in efficiency.

Findings

Shorter cooling schedule matching the optimal length conjectured.

Quantum algorithm achieves quadratic advantage in sample complexity.

Quadratic improvement in dependence on spectral gap.

Abstract

We present classical and quantum algorithms for approximating partition functions of classical Hamiltonians at a given temperature. Our work has two main contributions: first, we modify the classical algorithm of \v{S}tefankovi\v{c}, Vempala and Vigoda (\emph{J.~ACM}, 56(3), 2009) to improve its sample complexity; second, we quantize this new algorithm, improving upon the previously fastest quantum algorithm for this problem, due to Harrow and Wei (SODA 2020). The conventional approach to estimating partition functions requires approximating the means of Gibbs distributions at a set of inverse temperatures that form the so-called cooling schedule. The length of the cooling schedule directly affects the complexity of the algorithm. Combining our improved version of the algorithm of \v{S}tefankovi\v{c}, Vempala and Vigoda with the paired-product estimator of Huber (\emph{Ann.\ Appl.\ Probab.}, 25(2),~2015), our new quantum algorithm uses a shorter cooling schedule than previously known. This length matches the optimal length conjectured by \v{S}tefankovi\v{c}, Vempala and Vigoda. The quantum algorithm also achieves a quadratic advantage in the number of required quantum samples compared to the number of random samples drawn by the best classical algorithm, and its computational complexity has quadratically better dependence on the spectral gap of the Markov chains used to produce the quantum samples.